# 6. Series 30. Year

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### (3 points)1. heavy guns

Two machine guns, that are able to shoot bullets of mass $m=25g$ and speed $v_{1}=500\;\mathrm{m}\cdot \mathrm{s}^{-1}$ with at 10 rounds per second, are attached to the front of a car. The car accelerates on a flat surface to a speed $v_{2}=80\;\mathrm{km}\cdot h^{-1}$ and then starts firing. How many shots will be fired before the car stops? The car is neutral whilst shooting, the air and tyre resistance can be ignored. The heat losses in the machine guns are also negligible.

Mirek was thinking of GTA 2.

### (3 points)2. accidental drop

From what height would we need to „drop“ an object on a neutron star to make it land with a speed 0,1 $c$ (0,1 of speed of light). Our neutron star is 1.5 times heavier than our Sun and has diameter $d=10\;\mathrm{km}$. Ignore both the atmosphere of the star and its rotation. You can also ignore the correction for special relativity. However, do compare the results for a homogenous gravitational field (with the same strength as is on the star surface) and for a radial gravitational field.
**Bonus:** Do not ignore the special relativity correction.

Karel was thinking about neutron stars (yet again)

### (6 points)3. relativistic Zeno's paradox

Superman and Flash decided to race each other. The race takes place in deep space as there is no straight beach long enough on Earth. As Flash is slower, he starts with a length lead $l$ ahead of Superman. At one moment, Flash starts with a constant speed $v_{F}$ comparable with the speed of light. At the moment Superman sees that Flash started, he starts running at a constant speed $v_{S}>v_{F}$. How long will it take Superman to catch up with Flash (from Superman's point of view)? How long will it take from Flash's point of view? Was the starting method fair? Can you devise a more fair method (keeping the length lead $l)?$

### (7 points)4. shoot your rat

Mirek wants to shoot a rat he sees at the dorm. To that end, he made a simple air gun which can be modeled as a tube with constant cross-section $S=15\;\mathrm{mm}$ and length $l=30\;\mathrm{cm}$ closed on one side and open on the other. Mirek plans to place a bullet of mass $m=2g$ into the tube so that the bullet seals to tube exactly and is fixed at a distance $d=3\;\mathrm{cm}$ from the closed end. He that pumps up the closed section to a pressure $p_{0}$ and then releases the bullet. He wants the speed of the bullet to be at least $v=90\;\mathrm{m}\cdot \mathrm{s}^{-1}$ as it exits the tube. What pressure will he need to achieve if the gas is ideal? Discuss the realism of the situation. Assume the bullet is released by a quasi-static adiabatic process where $κ=7⁄5$, as the gas is diatomic. Assume an external atmospheric pressure $p_{a}=10^5Pa$. Neglect losses due to friction, air resistance and gas compression ahead of the bullet.

Karel wanted to find out if the solvers could pass the Masters programme admissions at MFF

### (8 points)5. hit him over the knuckles

Consider a homogeneous rod of constant cross-section and length $l$ attached to a freely rotating joint at one end. At the beginning, the rod points straight up and is in a homogeneous gravitational field with acceleration $g$. Due to a slight whiff of wind, the rod starts turning and „falling“ down, but is still held on the joint. Find the acceleration of the end of the rod in time.

### (9 points)P. evaporating asteroid

A very large piece of ice (let us say with diameter 1 km) is placed near a Sun-like star to a circular orbit. It is placed so close, that the equillibrium temperature of a black body at this distance would be approximately 30 ° C. What will happen with such an asteroid and its orbit? The asteroid is not tidally locked.

Karel likes astrophysics, so he came up with something again.

### (12 points)E. composition as if by Cimrman

Get a wine glass, ideally a thin one with a ground edge. First measure the internal diameter of the glass as a function of height from the bottom. Then use it to create sound by moving a wet finger along its edge (this requires pations). Measure how do the frequencies of tones created in this way depend on the height of water in the glass (measure at least 5 different heights and 2 frequencies at each height).
**Hint:** If the walls of the glass are thin, we can assume the internal dimensions are the same as external and measure the diameter as a function of height from an appropriate photograph with a scale. For measuring sound we recommend the free software Audacity (Analyze $→$ Plot spectrum).

Karel likes playing with glasses at formal dinners.

### (10 points)S. nonlinear

- Describe in your own words how and when the nonlinear regression can be used (it is sufficient to describe the following: nonlinear regression model, estimation of unknown regression coefficients, expression of uncertainty of estimates of regression coefficients and fitted values, statistical tests of hypotheses about regression coefficients, identifiability of regression coefficients and choice of regression function). It is not necessary to provide derivations and proofs, brief overview is sufficient.

- See values ($x_{i},y_{i})$ in the attached file
*regrese1.csv*. We want to fit the theoretical functional dependence, which in this case is a sinusoid, i.e. the function

$$f(x)=a b\cdot \sin(cx d)$$

Plot a graph of observed values and fitted function (such graph has to meet usual requirements) and provide brief interpretation. It is not necessary to do regression diagnostics.

**Hint:** Do not forget to correctly solve the identifiability problem of this model by suitable restrictive conditions on possible values of parameter $c$.

- See values ($x_{i},y_{i})$ in the attached file
*regrese2.csv*. We want to fit the theoretical functional dependence, which, in this case, is an exponential function, i.e. the function

$$f(x)=a \;\mathrm{e}^{bx c}$$

Provide values of estimates of all regression coefficients including corresponding standard errors.

**Hint:** Try to check (by the means of graphical methods) whether the assumption of homoskedasticity holds and if necessary, use White's estimate (sandwich estimate) of covariance matrix to compute standard errors correctly.

- See values ($x_{i},y_{i})$ in the attached file
*regrese3.csv*. We want to fit the theoretical functional dependence, which in this case is a hyperbolic function, i.e. the function

$$f(x)=a \frac{1}{bx c}$$

Plot a graph of observed values (in the form of error bars) and fitted function and provide brief interpretation. Perform regression diagnostics.

**Bonus:** See values ($x_{i},y_{i})$ in the attached file*regrese4.csv*. We want to fit the theoretical functional dependence, which in this case is too complicated to be expressed in analytical form. Try to fit regression splines (with suitably chosen knots and suitable degree). Plot a graph of observed values and fitted function.

It is recommended to use statistical software *R* for all computations. Sample *R* script (with comments in code explaining syntax of \emph{R} programming language) may be helpful (in Czech only). .

sample script in UTF-8 encoding (in Czech only)

sample script in CP-1250 encoding (in Czech only)

ukazka1.csv (data file used in sample script)

Michal thinks that the last round has to be as difficult as possible.